Native American is the mainstream term, American Indian is the distant second place but I've heard many use that term for themselves (eg the American Indian Movement or AIM).
I don't really see the term "Indian," as being completely incorrect, since in old European parlance, "India" was a generic term for all far-flung lands that were not China or Japan.
This is urban myth (perpetuated by the likes of Miriam-Webster). Eskimo does not mean "raw-meat-eater." It actually means "snowshoe-netter." But I think the myth has done enough damage that the actual people now do not like the term, and prefer their own tribal names.